The entire question of number 4a-4d

4) [8 points total] In class we analyzed the optimal income distribution under four important assumptions: Social welfare is strictly/additively utilitarian. Individuals have identical utility functions that depend only on income. . . The marginal utility of income diminishes with income. The total amount of income is fixed. Under those assumptions, it is socially optimal to transfer income from the rich to the poor until perfect income equality is achieved. In this problem you will assess how the socially optimal income distribution changes when you relax each of those assumptions. Consider a society with two people, Ebenezer and Bob. Social welfare is given by: W = UE + UB, where U is Ebenezer's utility and Up is Bob's utility. They have identical utility functions, where the marginal of utility of income diminishes with income: MUg = 3460 - 0.02/g and MUB = 3460 - 0.021B, where IF is Ebenezer's income and Ig is Bob's income. Finally, Ebenezer is endowed with $100,000 of wealth, whereas Bob has $0. Under the four assumptions listed above, half of Ebenezer's wealth should be taxed and transferred to Bob so that each has $50,000 of income. a. [2 points] Suppose that social welfare is not strictly utilitarian. Ebenezer is kind of a jerk, so society places twice as much weight on Bob's utility: W = Ug + 2 . Up. When you take a dollar from Ebenezer, social welfare decreases by MUg. But when you give that dollar to Bob, social welfare increases by 2 . MUg. What is the optimal distribution of income now? (Hint: The social welfare function has changed from what we considered in class. But there's still only $100,000 to go around; therefore, it is still the case that le + Ig = 100,000.) Page 1 of 3 b. [2 points] Go back to assuming that social welfare is strictly utilitarian. When you take a dollar from Ebenezer, social welfare decreases by MUE, and when you give that dollar to Bob, social welfare increases by MUg. However, Ebenezer likes money more than Bob. Ebenezer's marginal utility of income is still MU = 3460 - 0.02/g, but Bob's is lower and given by MUg = 2860 - 0.021g. What is the optimal distribution of income now? (Hint: Unlike the example in class, the two people have different marginal utility functions. But there's still only $100,000 to go around; therefore, it is still the case that Ig + Ig = 100,000.) c. [2 points] Go back to assuming that Ebenezer and Bob have the same utility functions. However, both have a constant (not diminishing) marginal utility of income given by MU = MUg = 3460. What is the optimal distribution of income now? (Hint: The two people have a constant marginal utility of income, so they amount of utility that they gain (or lose) with an increase (or decrease) in income doesn't depend on how much income they have. What does that imply about the change in social welfare when we take a dollar from Ebenezer and give it to Bob?) d. [2 points] Go back to assuming that Ebenezer and Bob have the same diminishing marginal utility function. (That is, MUE = 3460 - 0.02/F and MUg = 3460 -0.02/g.) However, the total amount of income is no longer fixed. Every $1.00 transferred to Bob costs Ebenezer $1.20, where the additional $0.20 of cost comes in the form of administrative costs. When you give a dollar to Bob, social welfare increases by MUg. But in order to get that dollar, $1.20 is taken from Ebenezer, which implies that social welfare decreases by 1.2 . MUF. What is the optimal distribution of income now? (Hint: There a total of $100,000 to go around only if we let Ebenezer keep everything. For every $1.00 that we